Question

question 11 (1 point)
suppose a function f(x) is differentiable everywhere and has a local minimum at x = c.
if f(x) < 0 when x < c, and f(x) > 0 when x > c, then by the global interval method we
know x = c is
a local minimum
an absolute maximum
an absolute minimum
a local maximum
view hint for question 11
level 2: advanced problems

Explanation:

According to the first - derivative test for local extrema: If a function \(y = f(x)\) is differentiable at \(x = c\), and \(f^{\prime}(x)\) changes sign from negative to positive as \(x\) increases through \(c\) (i.e., \(f^{\prime}(x)<0\) for \(x < c\) and \(f^{\prime}(x)>0\) for \(x>c\)), then \(x = c\) is a local minimum of the function \(y = f(x)\).
The global interval method is related to the first - derivative test. An absolute maximum/minimum is the largest/smallest value of the function over its entire domain (not just in a local neighborhood), and we have no information about the function's behavior over its entire domain. A local maximum would require \(f^{\prime}(x)>0\) for \(x < c\) and \(f^{\prime}(x)<0\) for \(x>c\).

Answer:

a local minimum